1 Pertemuan 12 Sampling dan Sebaran Sampling-2 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Menjelaskan Estimator dan sifat-sifatnya serta derajat kebebasan (degres of freedom)

3 Outline Materi Estimator dan Sifat-sifatnya Derajat Kebebasan (degrees of freedom)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., estimator An estimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the: Population Parameter Sample Statistic Mean (  ) is theMean (X) Variance (  2 )is the Variance (s 2 ) Standard Deviation (  )is the Standard Deviation (s) Proportion (p)is the Proportion ( ) estimator An estimator of a population parameter is a sample statistic used to estimate the parameter. The most commonly-used estimator of the: Population Parameter Sample Statistic Mean (  ) is theMean (X) Variance (  2 )is the Variance (s 2 ) Standard Deviation (  )is the Standard Deviation (s) Proportion (p)is the Proportion ( ) Desirable properties of estimators include: Unbiasedness Efficiency Consistency Sufficiency Desirable properties of estimators include: Unbiasedness Efficiency Consistency Sufficiency 5-4 Estimators and Their Properties

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., unbiased An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates. For example, E(X)=  so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean. systematic deviation bias Any systematic deviation of the estimator from the population parameter of interest is called a bias. unbiased An estimator is said to be unbiased if its expected value is equal to the population parameter it estimates. For example, E(X)=  so the sample mean is an unbiased estimator of the population mean. Unbiasedness is an average or long-run property. The mean of any single sample will probably not equal the population mean, but the average of the means of repeated independent samples from a population will equal the population mean. systematic deviation bias Any systematic deviation of the estimator from the population parameter of interest is called a bias. Unbiasedness

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., unbiased An unbiased estimator is on target on average. biased A biased estimator is off target on average. { Bias Unbiased and Biased Estimators

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., efficient An estimator is efficient if it has a relatively small variance (and standard deviation). efficient An efficient estimator is, on average, closer to the parameter being estimated.. inefficient An inefficient estimator is, on average, farther from the parameter being estimated. Efficiency

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., consistent An estimator is said to be consistent if its probability of being close to the parameter it estimates increases as the sample size increases. sufficient An estimator is said to be sufficient if it contains all the information in the data about the parameter it estimates. n = 100 n = 10 Consistency Consistency and Sufficiency

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., unbiased estimators efficient sufficient For a normal population, both the sample mean and sample median are unbiased estimators of the population mean, but the sample mean is both more efficient (because it has a smaller variance), and sufficient. Every observation in the sample is used in the calculation of the sample mean, but only the middle value is used to find the sample median. best In general, the sample mean is the best estimator of the population mean. The sample mean is the most efficient unbiased estimator of the population mean. It is also a consistent estimator. Properties of the Sample Mean

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., sample variance unbiased estimator The sample variance (the sum of the squared deviations from the sample mean divided by (n-1) is an unbiased estimator of the population variance. In contrast, the average squared deviation from the sample mean is a biased (though consistent) estimator of the population variance. EsE xx n E xx n () () () ()                       Properties of the Sample Variance

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Consider a sample of size n=4 containing the following data points: x 1 =10x 2 =12x 3 =16x 4 =? and for which the sample mean is: Given the values of three data points and the sample mean, the value of the fourth data point can be determined: 5-5 Degrees of Freedom x 4 = 14

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., If only two data points and the sample mean are known: x 1 =10x 2 =12x 3 =?x 4 =? The values of the remaining two data points cannot be uniquely determined: Degrees of Freedom (Continued)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., The number of degrees of freedom is equal to the total number of measurements (these are not always raw data points), less the total number of restrictions on the measurements. A restriction is a quantity computed from the measurements. The sample mean is a restriction on the sample measurements, so after calculating the sample mean there are only (n-1) degrees of freedom remaining with which to calculate the sample variance. The sample variance is based on only (n-1) free data points: The number of degrees of freedom is equal to the total number of measurements (these are not always raw data points), less the total number of restrictions on the measurements. A restriction is a quantity computed from the measurements. The sample mean is a restriction on the sample measurements, so after calculating the sample mean there are only (n-1) degrees of freedom remaining with which to calculate the sample variance. The sample variance is based on only (n-1) free data points: s xx n     () () Degrees of Freedom (Continued)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., A sample of size 10 is given below. We are to choose three different numbers from which the deviations are to be taken. The first number is to be used for the first five sample points; the second number is to be used for the next three sample points; and the third number is to be used for the last two sample points. Example 5-4 Sample # Sample Point i.What three numbers should we choose in order to minimize the SSD (sum of squared deviations from the mean).? Note:

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Solution: Solution: Choose the means of the corresponding sample points. These are: 83.6, 69.33, and ii.Calculate the SSD with chosen numbers. Solution: Solution: SSD = See table on next slide for calculations. iii.What is the df for the calculated SSD? Solution: Solution: df = 10 – 3 = 7. iv.Calculate an unbiased estimate of the population variance. Solution: Solution: An unbiased estimate of the population variance is SSD/df = /7 = Solution: Solution: Choose the means of the corresponding sample points. These are: 83.6, 69.33, and ii.Calculate the SSD with chosen numbers. Solution: Solution: SSD = See table on next slide for calculations. iii.What is the df for the calculated SSD? Solution: Solution: df = 10 – 3 = 7. iv.Calculate an unbiased estimate of the population variance. Solution: Solution: An unbiased estimate of the population variance is SSD/df = /7 = Example 5-4 (continued)

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Example 5-4 (continued) Sample #Sample PointMeanDeviationsDeviation Squared SSD SSD/df

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Using the Template

COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., Constructing a sampling distribution of the mean from a uniform population (n=10) using EXCEL (use RANDBETWEEN(0, 1) command to generate values to graph): Histogram of Sample Means Sample Means (Class Midpoints) Frequency CLASS MIDPOINTFREQUENCY Using the Computer

19 Penutup Dasar pengambilan keputusan adalah statistik yang diperoleh dari sampel (sample statistic) yang memiliki pola sebaran tertentu, oleh karena itu pengetahuan tentang sebaran penarikan sampel (sampling) ini sangat penting bagi pengambilan keputusan tersebut.